A092184 -id:A092184 - cp's OEIS frontend

A108741 Member r=100 of the family of Chebyshev sequences S_r(n) defined in A092184.

0, 1, 100, 9801, 960400, 94109401, 9221760900, 903638458801, 88547347201600, 8676736387298001, 850231618608002500, 83314021887196947001, 8163923913326692803600, 799981229484128697805801, 78389996565531285692164900, 7681419682192581869134354401, 752700738858307491889474566400

Offset: 0

Henry Bottomley, Jun 22 2005

Partial sums of A046173. [Joerg Arndt, Jun 10 2013]

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A000217, A004189.

I:=[0,1,100]; [n le 3 select I[n] else 99*Self(n-1)-99*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Feb 02 2016

LinearRecurrence[{99, -99, 1}, {0, 1, 100}, 20] (* Vincenzo Librandi, Feb 02 2016 *)

a(n) = ((49+20*sqrt(6))^n+(49-20*sqrt(6))^n -2)/96 = 98*a(n-1)-a(n-2)+2 = 99*a(n-1)-99*a(n-2)+a(n-3) = (a(n-1)-1)^2/a(n-2) = A004189(n)^2.
G.f.: -x*(x+1)/((x-1)*(x^2-98*x+1)). [Colin Barker, Oct 24 2012]
From Wolfdieter Lang, Feb 01 2016: (Start)
a(n) = (T(n, 49) - 1)/48 = (T(2*n, 5) - 1)/48 with Chebyshev's T polynomials A053120. See the name.
a(n) = A000217((T(n, 5) - 1)/2)/3. n >= 0.
a(n) = S(n-1, 10)^2 = A004189(n)^2, with Chebyshev's S polynomials A049310. This is the triangular number = 3*square number identity. Cf. the famous triangular number = square number identity: A000217((T(n, 3) - 1)/2) = S(n-1, 6)^2. A001109 and A001110. (End)

A098301 Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 16, 225, 3136, 43681, 608400, 8473921, 118026496, 1643897025, 22896531856, 318907548961, 4441809153600, 61866420601441, 861688079266576, 12001766689130625, 167163045568562176, 2328280871270739841, 32428769152221795600, 451674487259834398561

Offset: 0

Wolfdieter Lang, Oct 18 2004

Also m such that (3*m^2 + m)/4 = m*(3*m + 1)/4 is a perfect square. - Ctibor O. Zizka, Oct 15 2010

Consequently A049451(k) is a square if and only if k = a(n). - Bruno Berselli, Oct 14 2011

Colin Barker, Table of n, a(n) for n = 0..874
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A001075, A001353, A049451, A092184.

LinearRecurrence[{# - 1, -# + 1, 1}, {0, 1, #}, 20] &[16] (* Michael De Vlieger, Feb 23 2021 *)

concat(0, Vec(x*(1+x)/((1-x)*(1-14*x+x^2)) + O(x^50))) \\ Colin Barker, Jun 15 2015

a(n) = (T(n, 7)-1)/6 with Chebyshev's polynomials of the first kind evaluated at x=7: T(n, 7) = A011943(n) = ((7 + 4*sqrt(3))^n + (7 - 4*sqrt(3))^n)/2; therefore: a(n) = ((7 + 4*sqrt(3))^n + (7 - 4*sqrt(3))^n - 2)/12.
a(n) = A001353(n)^2 = S(n-1, 4)^2 with Chebyshev's polynomials of the second kind evaluated at x=4, S(n, 4):=U(n, 2).
a(n) = 14*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3), n >= 3.
G.f.: x*(1+x)/((1-x)*(1 - 14*x + x^2)) = x*(1+x)/(1 - 15*x + 15*x^2 - x^3) (from the Stephan link, see A092184).
Conjecture: 4*A007655(n+1) + A046184(n) = A055793(n+2) + a(n+1). - Creighton Dement, Nov 01 2004
a(n) = (A001075(n)^2-1)/3. - Parker Grootenhuis, Nov 28 2017

A098296 Member r=11 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 11, 100, 891, 7921, 70400, 625681, 5560731, 49420900, 439227371, 3903625441, 34693401600, 308336988961, 2740339499051, 24354718502500, 216452127023451, 1923714424708561, 17096977695353600, 151949084833473841

Offset: 0

Wolfdieter Lang, Oct 18 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-10,1).

a:=[0,1,11];; for n in [4..30] do a[n]:=10*a[n-1]-10*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

I:=[0,1,11]; [n le 3 select I[n] else 10*Self(n-1)-10*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019

LinearRecurrence[{10,-10,1},{0,1,11},30] (* Harvey P. Dale, Jan 27 2012 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+x)/((1-x)*(1-9*x+x^2)))) \\ G. C. Greubel, May 24 2019

(x*(1+x)/((1-x)*(1-9*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

a(n) = 2*(T(n, 9/2)-1)/7 with twice Chebyshev's polynomials of the first kind evaluated at x=9/2: 2*T(n, 9/2) = A056918(n) = ((9 + sqrt(77))^n + (9 - sqrt(77))^n)/2^n.
a(n) = 9*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 10*a(n-1) - 10*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=11.
G.f.: x*(1+x)/((1-x)*(1-9*x+x^2)) = x*(1+x)/(1-10*x+10*x^2-x^3) (from the Stephan link, see A092184).

A098297 Member r=12 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 12, 121, 1200, 11881, 117612, 1164241, 11524800, 114083761, 1129312812, 11179044361, 110661130800, 1095432263641, 10843661505612, 107341182792481, 1062568166419200, 10518340481399521, 104120836647576012

Offset: 0

Wolfdieter Lang, Oct 18 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A097784, A098296.

a:=[0,1,12];; for n in [4..30] do a[n]:=11*a[n-1]-11*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

I:=[0,1,12]; [n le 3 select I[n] else 11*Self(n-1)-11*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019

LinearRecurrence[{11,-11,1}, {0,1,12}, 30] (* G. C. Greubel, May 24 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+x)/((1-x)*(1-10*x+x^2)))) \\ G. C. Greubel, May 24 2019

(x*(1+x)/((1-x)*(1-10*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

a(n) = (T(n, 5)-1)/4 with Chebyshev's polynomials of the first kind evaluated at x=5: T(n, 5) = A001079(n) = ((5 + 2*sqrt(6))^n + (5 - 2*sqrt(6))^n)/2.
a(n) = 10*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=12.
G.f.: x*(1+x)/((1-x)*(1-10*x+x^2)) = x*(1+x)/(1-11*x+11*x^2-x^3) (from the Stephan link, see A092184).
a(n) = A132596(n) / 2. - Peter Bala, Dec 31 2012

A098306 Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 6, 49, 384, 3025, 23814, 187489, 1476096, 11621281, 91494150, 720331921, 5671161216, 44648957809, 351520501254, 2767515052225, 21788599916544, 171541284280129, 1350541674324486, 10632792110315761, 83711795208201600

Offset: 0

Wolfdieter Lang, Oct 18 2004

((-1)^(n+1))*a(n) = S_{-6}(n), n>=0, defined in A092184.

This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 6, P2 = -16, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to Chebyshev polynomials
Index entries for linear recurrences with constant coefficients, signature (7,7,-1).

Cf. A001090, A001091, A070997, A092184, A100047.

a[n_] := 1/10*((4 + Sqrt[15])^n + (4 - Sqrt[15])^n - 2*(-1)^n) // Simplify; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2017 *)
LinearRecurrence[{7, 7, -1}, {0, 1, 6, 49, 384, 3025}, 50] (* G. C. Greubel, Aug 08 2017 *)

x='x+O('x^50); Vec(x*(1-x)/((1+x)*(1-8*x+x^2))) \\ G. C. Greubel, Aug 08 2017

a(n) = (T(n, 4)-(-1)^n)/5, with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n)=((4+sqrt(15))^n + (4-sqrt(15))^n)/2.
a(n) = 8*a(n-1) - a(n-2) + 2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n) = 7*a(n-1) + 7*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=6.
G.f.: x*(1-x)/((1+x)*(1-8*x+x^2)) = x*(1-x)/(1-7*x-7*x^2+x^3) (from the Stephan link, see A092184).
From Peter Bala, Mar 25 2014: (Start)
a(2*n) = 6*A001090(n)^2; a(2*n+1) = A070997(n)^2.
a(n) = |u(n)|^2, where {u(n)} is the Lucas sequence in the quadratic integer ring Z[sqrt(-6)] defined by the recurrence u(0) = 0, u(1) = 1, u(n) = sqrt(-6)*u(n-1) - u(n-2) for n >= 2.
Equivalently, a(n) = U(n-1,sqrt(-6)/2)*U(n-1,-sqrt(-6)/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = 1/10*( (4 + sqrt(15))^n + (4 - sqrt(15))^n - 2*(-1)^n ).
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 4; 1, 3] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

A098308 Unsigned member r=-8 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 8, 81, 800, 7921, 78408, 776161, 7683200, 76055841, 752875208, 7452696241, 73774087200, 730288175761, 7229107670408, 71560788528321, 708378777612800, 7012226987599681, 69413891098384008, 687126683996240401

Offset: 0

Wolfdieter Lang, Oct 18 2004

((-1)^(n+1))*a(n) = S_{-8}(n), n>=0, defined in A092184.

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,9,-1).

LinearRecurrence[{9,9,-1},{0,1,8},40] (* Harvey P. Dale, Aug 11 2013 *)

a(n)= (T(n, 5)-(-1)^n)/6, with Chebyshev's polynomials of the first kind evaluated at x=5: T(n, 5)=A001079(n)=((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/2.
a(n)= 10*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 9*a(n-1) + 9*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=8.
G.f.: x*(1-x)/((1+x)*(1-10*x+x^2)) = x*(1-x)/(1-9*x-9*x^2+x^3) (from the Stephan link, see A092184).
a(x)=1/12(2*(-1)^x+(5+2*Sqrt[6])(5-2*Sqrt[6])^x+(5-2*Sqrt[6])(5+2*Sqrt[6])^x). - Harvey P. Dale, Aug 11 2013
a(n-1)+a(n) = A072256(n). - R. J. Mathar, Feb 19 2017

A098298 Member r=13 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 13, 144, 1573, 17161, 187200, 2042041, 22275253, 242985744, 2650567933, 28913261521, 315395308800, 3440435135281, 37529391179293, 409382867836944, 4465682155027093, 48713120837461081, 531378647057044800, 5796451996790031721, 63229593317633304133

Offset: 0

Wolfdieter Lang, Oct 18 2004

Colin Barker, Table of n, a(n) for n = 0..950
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,-12,1).

Cf. A098296, A098297.

a:=[0,1,13];; for n in [4..30] do a[n]:=12*a[n-1]-12*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

[n le 2 select n-1 else 11*Self(n-1)- Self(n-2) + 2: n in [1..30]]; // Vincenzo Librandi, Mar 06 2016

LinearRecurrence[{12,-12,1},{0,1,13},30] (* Harvey P. Dale, May 11 2012 *)
RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 11 a[n-1] - a[n-2] + 2}, a, {n, 30}] (* Vincenzo Librandi, Mar 06 2016 *)

concat(0, Vec(x*(1+x)/((1-x)*(1-11*x+x^2)) + O(x^25))) \\ Colin Barker, Mar 06 2016

(x*(1+x)/((1-x)*(1-11*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

a(n) = 2*(T(n, 11/2) - 1)/9 with twice Chebyshev's polynomials of the first kind evaluated at x=11/2: 2*T(n, 11/2) = A057076(n) = ((11 + sqrt(117))^n + (11 - sqrt(117))^n)/2^n.
a(n) = 11*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 12*a(n-1) - 12*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=13.
G.f.: x*(1+x)/((1-x)*(1-11*x+x^2)) = x*(1+x)/(1-12*x+12*x^2-x^3) (from the Stephan link, see A092184).

A098299 Member r=14 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 14, 169, 2016, 24025, 286286, 3411409, 40650624, 484396081, 5772102350, 68780832121, 819597883104, 9766393765129, 116377127298446, 1386759133816225, 16524732478496256, 196910030608138849

Offset: 0

Wolfdieter Lang, Oct 18 2004

Michael De Vlieger, Table of n, a(n) for n = 0..930
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (13,-13,1).

LinearRecurrence[{13, -13, 1}, {0, 1, 14}, 18] (* Michael De Vlieger, Feb 23 2021 *)

a(n) = (T(n, 6)-1)/5 with Chebyshev's polynomials of the first kind evaluated at x=6: T(n, 6)=A023038(n)= ((6+sqrt(35))^n + (6-sqrt(35))^n)/2.
a(n) = 12*a(n-1) - a(n-2) + 2, n>=2, a(0)=0, a(1)=1.
a(n) = 13*a(n-1) - 13*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=14.
G.f.: x*(1+x)/((1-x)*(1-12*x+x^2)) = x*(1+x)/(1-13*x+13*x^2-x^3) (from the Stephan link, see A092184).

A098300 Member r=15 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 15, 196, 2535, 32761, 423360, 5470921, 70698615, 913611076, 11806245375, 152567578801, 1971572279040, 25477872048721, 329240764354335, 4254652064557636, 54981236074894935, 710501416909076521

Offset: 0

Wolfdieter Lang, Oct 18 2004

Michael De Vlieger, Table of n, a(n) for n = 0..900
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (14,-14,1).

LinearRecurrence[{# - 1, -# + 1, 1}, {0, 1, #}, 18] &[15] (* Michael De Vlieger, Feb 23 2021 *)

a(n) = 2*(T(n, 13/2)-1)/11 with twice the Chebyshev polynomials of the first kind evaluated at x=13/2: 2*T(n, 13/2)=A078363(n)=((13+sqrt(165))^n + (13-sqrt(165))^n)/2^n.
a(n) = 13*a(n-1) - a(n-2) + 2, n>=2, a(0)=0, a(1)=1.
a(n) = 14*a(n-1) - 14*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=15.
G.f.: x*(1+x)/((1-x)*(1-13*x+x^2)) = x*(1+x)/(1-14*x+14*x^2-x^3) (from the Stephan link, see A092184).

A098302 Member r=17 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 17, 256, 3825, 57121, 852992, 12737761, 190213425, 2840463616, 42416740817, 633410648641, 9458742988800, 141247734183361, 2109257269761617, 31497611312240896, 470354912413851825, 7023826074895536481

Offset: 0

Wolfdieter Lang, Oct 18 2004

Michael De Vlieger, Table of n, a(n) for n = 0..852
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,-16,1).

LinearRecurrence[{# - 1, -# + 1, 1}, {0, 1, #}, 18] &[17] (* Michael De Vlieger, Feb 23 2021 *)

a(n) = 2*(T(n, 15/2)-1)/13 with twice the Chebyshev polynomials of the first kind evaluated at x=15/2: 2*T(n, 15/2)=A078365(n)= ((15+sqrt(221))^n +(15-sqrt(221))^n)/2^n.
a(n) = 15*a(n-1) - a(n-2) + 2, n>=2, a(0)=0, a(1)=1.
a(n) = 16*a(n-1) - 16*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=17.
G.f.: x*(1+x)/((1-x)*(1-15*x+x^2)) = x*(1+x)/(1-16*x+16*x^2-x^3) (from the Stephan link, see A092184).
a(2*n+2) = 17*A078364(n)^2; a(2*n+1) = A161591(n+1)^2. - Klaus Purath, Aug 13 2025

cp's OEIS Frontend

A108741 Member r=100 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A098301 Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A098296 Member r=11 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A098297 Member r=12 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A098306 Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A098308 Unsigned member r=-8 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A098298 Member r=13 of the family of Chebyshev sequences S_r(n) defined in A092184.

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